A particle of mass (m) is executing oscillations about the origin on the (x) axis. Its potential energy is `V(x) = k|x|^3` where (k) is a positive constant. If the amplitude of oscillation is a, then its time period (T) is.

Options

A. proportional to `(1)/(sqrta)`
B. independent of a
C. proportional to `sqrta`
D. proportional to `a^(3//2)`

Answer

A

Explanation

`U(x)=k|x|^3`
`F=-(dU)/(dx)=-3k|x|^2` (i)
For SHM `F=-momega^2x` (ii)
Comparing (i) and (ii)
`omega=sqrt((3kx)/(m))`
`T=(2pi)/(omega)=2pisqrt((m)/(3kx))=2pisqrt((m)/(3kasinomegat))`
`Tprop1/sqrta`